Given below are two statements: One is labelled as Assertion (A) and other is labelled as Reason (R) Assertion (A) : Time period of oscillation of a liquid drop depends on surface tension S , if densi

Question

Given below are two statements: One is labelled as Assertion (A) and other is labelled as Reason (R)

Assertion (A) : Time period of oscillation of a liquid drop depends on surface tension $(S)$ , if density of the liquid is $ρ$ and radius of the drop is $r$ , then $T = K \sqrt{\frac{ρ r^{3}}{S^{\frac{3}{2}}}}$ is dimensionally correct, where $K$ is dimensionless.

Reason (R) : Using dimensional analysis we get R.H.S. having different dimension than that of time period. In the light of above statements, choose the correct answer from the options given below.

TestHub · Accepted Answer

Using dimensional analysis for the given equation,
$T = K \sqrt{\frac{ρ r^{3}}{S^{\frac{3}{2}}}}$
$\Rightarrow [K] = \frac{[T] [S^{\frac{3}{4}}]}{[ρ^{\frac{1}{2}}] [r^{\frac{3}{2}}]} \Rightarrow [K] = \frac{[T] [M^{\frac{3}{4}} T^{- \frac{3}{2}}]}{[M^{\frac{1}{2}} L^{- \frac{3}{2}}] [L^{\frac{3}{2}}]} \Rightarrow [K] = [M^{\frac{1}{4}} T^{- \frac{1}{2}}]$
Therefore, $K$ is not dimensionless. Hence, Assertion is false.
For dimensional validity of the given equation, $K$ can not be dimensionless. If it is dimensionless then
Dimensions of L.H.S $\neq$ Dimensions of R.H.S
$∴$ Option (D) is correct.

Physics - Unit & Dimension Question with Solution | TestHub

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